(a) writ the expression for y as a function of x and t for a sinusoidal wave travelling along a rope in the neagative x direction with the folllowing characteristics: A =8.00 cm , `delta=80.0 cm,f=3.00 Hz`, and `y(0,t)=0 at t=0`, (b) write an expression for y as a function of x and t for the wave in part (a) assuming that y(x,0)=0 at the point x=10.0 cm.

Explanation

Think about the graph of `y` as a function of `x` at one instant as a smoooth succession of identical crests and troughs, with the length in space of each cycle being `80.0 cm`. the distance from the top of a crest to the bottom of a trough is `16.0 cm`. now think of the whole graph moving towards the left at 240 `cm//s`.
using the travelling wave model, we can put constant with the right values into `y=A sin (kx+omegat+phi)` to have the mathetical representation of the wave. we have the same (positive) sings for both kx and omega `t` so that a point of constant phase will be at a decreasing value of `x` as t increase that is, so that the wave will move will move to the left.
The amplitude is `A=y_(max) =8.00 cm=0.0800 m`
The wave number is
`k=(2pi)/lambda=(2pi)/(0.800 m)=(5x)/(2)`
the angular frequency, `omega=2pif=2pi(3.00 s^(-1)=6.00pi rad//s`
(a) we have `y=A sin (ks+omegat+phi)`, choosing `phi=0` will make it true that `y(0,0)=0`. then the wave function becomes upon sustitution of the constant value for this wave
`y=(0.080 m)sin((5pi)/(2)x+6.00pit)`
(b) in general, `y=(0.0800 m)sin((5pi)/(2)x+6.00pit+phi)` If `y(x,0)=0 at x=0.100 m`, we require
`0=(0.0800 m)sin((5)/(2)pixx0.1+phi)`
`(5pi)/(2)xx0.1+phi=0`
so we must have the phase constant `phi=-(pi)/(4) rad` therefore, the wave function for all values of the variable `x` and `t` is
`ypi(0.0800 m)sin((5)/(2)pixpi6.00pi t-(pi)/(4))`